Assignment 1
Reading Assignment:
1. Chapter 1: Functions (From the course notes posted on piazza)2. Chapter 2: Metric Spaces and Topology (both from Moon’s book and course notes)
Problems:
1. Let
x
= (
x
1
,...,x
n
)
,y
= (
y
1
,...,y
n
)
∈
R
n
and consider the function
ρ
given by
ρ
x,y
= max
{
x
1
−
y
1

,...,

x
n
−
y
n
}
.
Show that
ρ
is a metric.2. Let
X
be a metric space with metric
d
. Deﬁne ¯
d
:
X
×
X
→
R
by¯
d
(
x,y
) = min
{
d
(
x,y
)
,
1
}
.
Show that ¯
d
is also a metric.3. The image of a function applied to a setvalued argument is deﬁned by
f
(
A
)
{
f
(
x
)

x
∈
A
}
and
f
−
1
(
B
)
{
x
∈
X

f
(
x
)
∈
B
}
. Let
f
:
X
→
Y
,
A
⊆
X
, and
B
⊆
Y
.(a) Show that
f
−
1
(
f
(
A
))
⊇
A
and that equality holds if
f
is injective.(b) Show that
f
(
f
−
1
(
B
))
⊆
B
and that equality holds if
f
is surjective.4. Let
ℓ
2
denote the subset of
R
ω
consisting of all sequences (
x
1
,x
2
,...
) such that
i
x
2
i
converges. (You may assume the standard facts about inﬁnite series.)(a) Show that if
x
,
y
∈
ℓ
2
, then
i

x
i
y
i

converges.(b) Let
c
∈
R
. Show that if
x
,
y
∈
ℓ
2
, then so are
x
+
y
and
c
x
.(c) Show that
d
(
x
,
y
) =
∞
i
=1
(
x
i
−
y
i
)
2
1
/
2
is a welldeﬁne metric on
ℓ
2
. It is called the
ℓ
2
metric.5. Let
x
n
→
x
and
y
n
→
y
in the space
R
with metric
d
(
x,x
′
) =

x
−
x
′

. Show that
x
n
+
y
n
→
x
+
yx
n
−
y
n
→
x
−
yx
n
y
n
→
xy,
and provided that each
y
n
= 0 and
y
= 0,
x
n
/y
n
→
x/y.
[Hint: First show that +
,
−
,
·
,/
are continuous functions from (
R
2
,d
1
) to (
R
,
·
).]1
6. Deﬁne
f
n
: [0
,
1]
→
R
by the equation
f
n
(
x
) =
x
n
. Show that the sequence
{
f
n
(
x
)
}
convergesfor each
x
∈
[0
,
1], but that the sequence
{
f
n
}
does not converge uniformly. Recall thatuniform convergence to
f
on [
a,b
] implies that, for any
ǫ >
0, there exists an
N
such that

f
n
(
t
)
−
f
(
t
)

< ǫ
for all
n > N
and all
t
∈
[
a,b
].
Optional Problems:
1. Show that
d
′
(
x,y
) =
d
(
x,y
)
/
(1 +
d
(
x,y
)) is a bounded metric for
X
if
d
is a metric for
X
.2. Prove continuity of the algebraic operations on
R
, as follows: Use the metric
d
(
a,b
) =

a
−
b

on
R
and the metric
ρ
((
x,y
)
,
(
x
0
,y
0
)) = max
{
x
−
x
0

,

y
−
y
0
}
on
R
2
.(a) Show that addition is continuous. [Hint: Given
ǫ
, let
δ
=
ǫ/
2 and note that
d
(
x
+
y,x
0
+
y
0
)
≤
x
−
x
0

+

y
−
y
0

.
](b) Show that multiplication is continuous. [Hint: Given (
x
0
,y
0
) and
ǫ >
0, let3
δ
= min
{
ǫ/
(

x
0

+

y
0

+ 1)
,
√
ǫ
}
and note that
d
(
xy,x
0
y
0
)
≤
x
0

y
−
y
0

+

y
0

x
−
x
0

+

x
−
x
0

y
−
y
0

.
](c) Show that the operation of taking reciprocals is a continuous map from
R
−{
0
}
to
R
.[Hint: Given
x
0
= 0 and
ǫ >
0, let
δ
= min
{
x
0

/
2
,x
20
ǫ/
2
}
. Note that
d
(1
/x,
1
/x
0
) =

x
−
x
0

/

xx
0

. If

x
−
x
0

< δ
, then

xx
0
−
x
20

<

x
0

2
/
2, so
xx
0
−
x
20
>
−
x
20
/
2 and
xx
0
> x
20
/
2
>
0.](d) Show that the subtraction and quotient opertions are continuous.2